Connecting orbits in highdimensional dynamical systems
31/00 Wolf-Jürgen Beyn, Thorsten Pampel (Göke), W. Semmler.
Dynamic optimization and Skiba sets in economic
examples
We discuss two optimization problems
from economics. The first is a model of optimal investment and
the second is a model of resource management. In both cases the
time horizon is infinite and the optimal control variables are
continuous. Typically, in these optimal control problems
multiple steady states and periodic orbits occur. This leads to
multiple solutions of the state-costate system each of which
relates to a locally optimal strategy but has its own limiting
behavior (stationary or periodic). Initial states that allow
different optimal solutions with the same value of the
objective function are called Skiba points. The set of Skiba
points is of interest, because it provides thresholds for a
global change of optimal strategies. We provide a systematic
numerical method for calculating locally optimal solutions and
Skiba points via boundary value problems. In parametric or
higher dimensional systems Skiba curves (or manifolds) appear
and we show how to follow them by a continuation process. We
apply our method to the models above where Skiba sets consist
of points or curves and where optimal solutions have different
stationary or periodic asymptotic behavior.
